Independence proof concerning compact cardinals, continuum hypotbesis, and normal ultrafilters

我们研究了K<@ d1 +@>D1-C.C. []SY。di-substituted正确。[) k -策略闭合强迫加入任意数量的k的闭合无界子集a彩票准备之后，这种强迫保留了k的强紧性。在得到的模型中，所有常规超滤波器对地面模型的限制是一致的。我们知道一个对称模型，在这个模型中广义连续统假设在强紧性基数K以下成立，在K处失效。然而，结果证明，将同样的东西与选择公理一起建立起来比预期的要困难得多。当我们在对称模型上强制选择轴子时，不清楚K的强紧性是否保留。以下是一些事实和新技术的发现。强制：(1)在P_klambda上添加许多闭合无界集(2)在P_klambda上添加非反射平稳集(3)在P_kK^<+++>上使用K超紧强制一个非正则超滤波器。组合学：(1)P_klambda中pii -不可描述性的组合表征。(2)有界理想可能具有也可能不具有划分性质。(3)有界理想并不突出性，期望进一步的研究使P_klambda的分拆性与不可言说性之间的关系变得清晰，从而在P_klambda的组合学上取得更大的进展。

We investigated a K<@D1+@>D1-C.C.(]SY.di-substituted right.[)K-strategically closed forcing adding arbitrary nunber of closed unbounded subsets of K.A Lottery preparation followed by this forcing preserves the strong compactness of K.In the resulting model the restricions of all normal ultrafilters to the ground model coincide.We know a symmetric model in which the genemlized continuum hypothesis holds below a strongly compact cardinal K and fails at K.However it turned out establishing the same thing together with the axiom of choice is far more difficult than had been expected. It is not clear whether the strong compactness of K is preserved when we force the the axion of choice on the symmetric model.Several facts and new technique are founded as in the folllowing.Forcing : (1)Adding many closed unbounded sets (2)Adding non-reflecting stationary sets to P_klambda (3)Forcing a non-regular ultrafilter on P_kK^<+++> with K supercompact.Combinatorics : (1)Combinatorial characterization of PIi-indescribability in P_klambda.(2)Bounded ideal may or may not have the partition property.(3)Bounded ideal is not precipitousIt is expected that further research makes the relation between the partition property and ineffability of P_klambda clear to bring great progress in the combinatorics of P_klambda.